Generalized Cattaneo (telegrapher's) equations in modeling anomalous diffusion phenomena

Fisher and Shannon Functionals for Hyperbolic Diffusion

The complexity measure for the distribution in space-time of a finite-velocity diffusion process is calculated. Numerical results are presented for the calculation of Fisher's information, Shannon's entropy, and the Cramér-Rao inequality, all of which are associated with a positively normalized solution to the telegrapher's equation. In the framework of hyperbolic diffusion, the non-local Fisher's information with the x-parameter is related to the local Fisher's information with the t-parameter. A perturbation theory is presented to calculate Shannon's entropy of the telegrapher's equation at long times, as well as a toy model to describe the system as an attenuated wave in the ballistic regime (short times).

Penetration of waves in global stochastic conducting media

The attenuation in the propagation of a plane wave in conducting media has been studied. We analyzed a wave motion suffering dissipation by the Joule effect in its propagation in a medium with global disorder. We solved the stochastic telegrapher's equation in the Fourier-Laplace representation allowing us to find the space penetration length of a plane wave in a complex conducting medium. Considering fluctuations in the loss of energy, we found a critical value k_{c} for Fourier's modes, thus if |k|<k_{c} the waves are localized. We showed that the penetration length is inversely proportional to k_{c}. Thus, the penetration length L=k_{c}^{-1} becomes an important piece of information for describing wave propagation with Markovian and non-Markovian fluctuations in the rate of the absorption of energy τ^{-1}. In addition, intermittent fluctuations in this rate have also been studied.

Accurate analytical calculation of the rate coefficient for the diffusion-controlled reactions due to hyperbolic diffusion

Using an approach based on the diffusion analog of the Cattaneo-Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski's boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski's theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice's formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper bound of the exact value. Here, the obtained formula seems to be of great significance for bridging a known gap between an analytically estimated rate coefficient on the one hand and molecular dynamics simulations together with experimentally observed results for the short times regime on the other hand. A particular emphasis has been placed on the rigorous mathematical treatment and important properties of the relevant initial boundary value problems in parabolic and hyperbolic diffusion theories.

Stochastic telegrapher's approach for solving the random Boltzmann-Lorentz gas

The 1D random Boltzmann-Lorentz equation has been connected with a set of stochastic hyperbolic equations. Therefore, the study of the Boltzmann-Lorentz gas with disordered scattering centers has been transformed into the analysis of a set of stochastic telegrapher's equations. For global binary disorder (Markovian and non-Markovian) exact analytical results for the second moment, the velocity autocorrelation function, and the self-diffusion coefficient are presented. We have demonstrated that time-fluctuations in the lost of energy in the telegrapher's equation, can delay the entrance to the diffusive regime, this issue has been characterized by a timescale t_{c} which is a function of disorder parameters. Indeed, producing a longer ballistic dynamics in the transport process. In addition, fluctuations of the space probability distribution have been studied, showing that the mean value of a stochastic telegrapher's Fourier mode is a good statistical object to characterize the solution of the random Boltzmann-Lorentz gas. In a different context, the stochastic telegrapher's equation has also been related to the run-and-tumble model in Biophysics. Then a discussion devoted to the potential applications when swimmers' speed and tumbling rate have time fluctuations has been pointed out.

Localization of plane waves in the stochastic telegrapher's equation

From the exact solution of the stochastic telegrapher's equation, Fourier plane-wave-like modes are introduced. Then the time evolution of the plane-wave modes are analyzed when the absorption of energy in the telegrapher's equation has strong time fluctuations. We demonstrate that fluctuations in the loss of energy introduce a localized gap with a size that depends on the correlation timescale of the fluctuations. We prove that for a large time correlation the gap is strongly reduced, which means that there is delocalization in the plane-wave modes with respect to the plane waves in the ordinary telegrapher's equation. This result is of relevance in the study of the transport of electromagnetic waves in a conducting medium, and sheds light on the functional role of the fluctuations in the loss of energy in the telegrapher's dynamics.

Fractional telegraph equation under moving time-harmonic impact

The time-fractional telegraph equation with moving time-harmonic source is considered on a real line. We investigate two characteristic versions of this equation: the "wave-type" with the second and Caputo fractional time-derivatives as well as the "heat-type" with the first and Caputo fractional time-derivatives. In both cases the order of fractional derivative 1 < α < 2. For the time-fractional telegraph equation it is impossible to consider the quasi-steady-state corresponding to the solution being a product of a function of the spatial coordinate and the time-harmonic term. The considered problem is solved using the integral transforms technique. The solution to the "wave-type" equation contains wave fronts and describes the Doppler effect contrary to the solution for the "heat-type" equation. Numerical results are illustrated graphically for different values of nondimensional parameters.

Integral decomposition for the solutions of the generalized Cattaneo equation

We present the integral decomposition for the fundamental solution of the generalized Cattaneo equation with both time derivatives smeared through convoluting them with some memory kernels. For power-law kernels t^{-α}, α∈(0,1] this equation becomes the time fractional one governed by the Caputo derivatives in which the highest order is 2. To invert the solutions from the Fourier-Laplace domain to the space-time domain we use analytic methods based on the Efross theorem and find out that solutions looked for are represented by integral decompositions which tangle the fundamental solution of the standard Cattaneo equation with nonnegative and normalizable functions being uniquely dependent on the memory kernels. Furthermore, the use of methodology arising from the theory of complete Bernstein functions allows us to assign such constructed integral decompositions the interpretation of subordination. This fact is preserved in two limit cases built into the generalized Cattaneo equations, i.e., either the diffusion or the wave equations. We point out that applying the Efross theorem enables us to go beyond the standard approach which usually leads to the integral decompositions involving the Gaussian distribution describing the Brownian motion. Our approach clarifies puzzling situation which takes place for the power-law kernels t^{-α} for which the subordination based on the Brownian motion does not work if α∈(1/2,1].

Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach

In recent years, different experimental works with molecular simulation techniques have been developed to study the transport of plasma-generated reactive species in liquid layers. Here, we improve the classical transport model that describes the molecular species movement in liquid layers via considering the fractional reaction–telegraph equation. We have considered the fractional equation to describe a non-Brownian motion of molecular species in a liquid layer, which have different diffusivities. The analytical solution of the fractional reaction–telegraph equation, which is defined in terms of the Caputo fractional derivative, is obtained by using the Laplace–Fourier technique. The profiles of species density with the mean square displacement are discussed in each case for different values of the time-fractional order and relaxation time.